Rayleigh-Taylor instability at a tilted interface in incompressible laboratory experiments and compressible numerical simulations Joanne M. Holford † Stuart B. Dalziel † David Youngs ‡ † DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK ‡ AWE Aldermaston, Reading, Berkshire, RG7 4PR, UK DAMTP, University of Cambridge Rayleigh-Taylor instability
Outline • Introduction RT instability at a tilted interface Mixing, available energy and mixing efficiency • Laboratory experiments At DAMTP, in the Fluid Dynamics Laboratory Incompressible water, NaCl to create density contrast • Numerical simulations At AWE, using Turmoil3D (with David Youngs) Compressible code, for a mixture of two ideal gases • Conclusions and further work DAMTP, University of Cambridge Rayleigh-Taylor instability
Introduction • RT instability Instability of dense fluid accelerated into less dense fluid ρ 1 ρ 1 ρ 1 g ρ 1 > ρ 2 ρ 2 ρ 2 ρ 2 ρ − ρ ′ g Non-dimensional parameter Atwood number A = = 1 2 g ρ + ρ 2 1 2 H For an external lengthscale H , timescale τ = Ag Much more efficient mixing than other mechanisms (shear instability, mechanical stirring) An important mixing process within larger-scale flows (3D instability of 2D shear billows) In environment, R-T instability has non-ideal initial conditions At a tilted interface, there is competition between local instability and large-scale overturning DAMTP, University of Cambridge Rayleigh-Taylor instability
• Definitions of mixing Distinguish between reversible and irreversible mixing: Reversible mixing - interleaving of fluid with different properties - “reversible mixing = stirring” Irreversible mixing - homogenisation of fluid properties at the molecular scale - “irreversible mixing = stirring + diffusion” Irreversible mixing is important for • chemical reactions • removal of available energy when mixing density gradients across a gravitational field • How do we measure mixing? Mixing can be measured by a molecular mixing fraction For two fluids, volume fractions f and (1- f ): ϑ ( , ) = − x t f x t f x t ( , )( 1 ( , )) Alternatively, for fluids of varying density in a gravitational field, can measure the mixing efficiency η For a fluid at rest, stirred by an energy input and returning to rest, η= increase in potential energy amount of energy added fraction of energy lost to fluid motion doing work against gravity DAMTP, University of Cambridge Rayleigh-Taylor instability
• Mixing in R-T instability Measurements of η in laboratory experiments - high values with some dependence on A Linden & Redondo (1991) Numerical simulations show sensitivity to initial conditions Linden, Redondo & Youngs (1991), Cook & Dimotakis (2001) • Diffusion and viscosity in incompressible fluids Mechanical energy density per unit volume E v = ρ + ρ 1 u gz 2 2 ∂ ( ) ( ) ( ) +∇ =− ε t E x t f x t x t , , . , , ∂ v v v f v energy flux ε v energy dissipation Water/salt system - ν = 1.0 × 10 -2 cm 2 s -1 kinematic viscosity κ = 1.4 × 10 -5 cm 2 s -1 diffusivity concentration fluctuations persist at smaller scales than ν velocity fluctuations - Pr = = 700 κ Turbulent flows - eddy viscosity = eddy diffusivity effectively Pr = 1 DAMTP, University of Cambridge Rayleigh-Taylor instability
• Available energy in incompressible flow = ∫ Mechanical energy in whole fluid E Ev dV decomposes: V + + PE back PE avail KE ��� ������� E back E avail Lorenz (1955), Thorpe (1977), Winters et al. (1995) In unforced, decaying flow d loss of E due to + = − ε dt E E back avail turbulent dissipation d ( ) gain in E back due to back = dt E q molecular mixing Define cumulative mixing efficiency t ∫ q dt ∆ PEback t η = = 0 cumulative t − ∆ Eavail ∫ + ε q dt t 0 and instantaneous mixing efficiency δ PE q η instantaneous = + = back ε −δ q E avail DAMTP, University of Cambridge Rayleigh-Taylor instability
• Available energy in compressible flow Now concerned with total energy (mechanical + internal) so 1 2 ρ , e internal energy. In whole fluid: = ρ + ρ + Ev u gz e 2 + + + + PE back IE back PE avail IE avail KE ����� ������� E back E avail Lorenz (1955), Andrews (1981), Shepherd (1993) In unforced, decaying flow d ( ) E is conserved + =0 dt E E back avail d ( ) gain in IE back due to dt IE back = ε turbulent dissipation and molecular mixing d ( ) gain in PE back due to back = dt PE q molecular mixing Same definitions of mixing efficiency apply, so η is still fraction of energy lost to fluid motion (reduction in E avail ) doing work against gravity (gain in E back ) DAMTP, University of Cambridge Rayleigh-Taylor instability
Laboratory Experiments • Configuration W =20cm H =50cm plane of measurement: L =40cm 5mm light sheet • Initial conditions A solid barrier introduces significant shear Reduced shear barrier: Dalziel, Linden & Youngs (1999) tank interior SIDE VIEW fabric Removal of finite thickness barrier causes initial velocity field DAMTP, University of Cambridge Rayleigh-Taylor instability
• Diagnostic measurements Image analysis: spatial resolution 1 pixel ≅ 0.1cm temporal resolution 25Hz Assume statistical homogeneity across tank Add propanol to fresh water to match refractive index Density measurement Dense fluid dyed with fluorescent dye Images corrected for divergence of light sheet and attenuation Velocity measurement Fluid seeded with 400 µ m neutrally-buoyant particles Lagrangian tracks for particles from tracking a frame sequence Interpolating onto a grid gives Eulerian velocities Gridded at two scales: 1cm - resolved velocity 3cm - mean velocity (overcomes lack of similarity between experiments) Assume isotropy at small scales ⇒ estimate of total KE • Parameters Atwood number 0.5 × 10 -3 < A < 2.5 × 10 -3 ⇒ Boussinesq Timescale 10s > τ > 4.5s RMS velocity 0.8cms -1 < u < 2cms -1 Integral lengthscale 1.8cm < l < 2.5cm Reynolds number 150 < Re < 500 DAMTP, University of Cambridge Rayleigh-Taylor instability
corrected for by comparing fluorescence divergence pattern of image with uniform dye concentration light source corrected for by integrating attenuation along rays to determine the (of incident, actual illumination not fluoresced and fluorescence light) at each point light rays DAMTP, University of Cambridge Rayleigh-Taylor instability
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Numerical Simulations • Code type Semi-Lagrangian finite volume code (conservation of fluid masses and momentum) Two ideal gases ( γ = 5/3) Typical simulation: 3D at resolution 200 × 160 × 80 • Viscosity and diffusion Loss of resolution at grid scale ⇒ diffusion-like behaviour for mass fractions, analogous to molecular mixing for KE, analogous to dissipation, and added to IE In some runs, an explicit viscosity was added • Approximating an incompressible fluid Normalisation: choose H = 1, Ag = 1, ρ 1 = 1 Non-dimensional parameters (ideally small): ∆ρ 2 = = ≈ 018 B Density ratio . ρ 0 g ρ AgH 3 = ≈ ≈ 008 M Mach number . γ p p 5 0 ρ 2 gH g 2 = ≈ ≈ 012 I Incompressibility ratio . ∆ρ p p 10 5 0 Compromise g = 11, p 0 = 100 DAMTP, University of Cambridge Rayleigh-Taylor instability
• Initial conditions - basic distribution Away from interface: Since u ≈ 0 , require ∂ p ∂ = − ρ . g z Require neutral stability, buoyancy frequency g ∂ g T ⇔ isentropic fluid p = k ( s ) ρ γ . = ∂ + = N 2 0 c p z T At interface: Choose specific heats at constant volume, c v 1 and c v 2 . Require temperature continuous ⇔ c v 1 ρ 1 = c v 2 ρ 2 . Everywhere: Pressure field cannot be entirely hydrostatic. ∂ ( ) Require ∂ t ∇ = . u 0 . Ignoring terms of O( u 2 ) , require 1 ∇ ∇ ∝∇ 1 γ ∇ γ −1 γ = p k p / ( ) / , . . 0 ρ with ∂ p = − ρ n z on boundaries with outward normal � n . g � . � ∂ n • Initial conditions - perturbations 2D velocity field with vorticity at interface models experimental barrier withdrawal. 3D random perturbation to interface position, wavelengths L L H < λ < , rms amplitude σ = 2500 . 40 20 DAMTP, University of Cambridge Rayleigh-Taylor instability
DAMTP, University of Cambridge Rayleigh-Taylor instability
DAMTP, University of Cambridge Rayleigh-Taylor instability
Energy budget Numerical results η ≈ 0.48 Typical experimental results η ≈ 0.38 1.5 1.0 Energy components E ( t )/ PE (0) 0.5 0.0 -0.5 -1.0 0 1 2 3 4 5 6 7 8 9 10 Time t / τ DAMTP, University of Cambridge Rayleigh-Taylor instability
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